Abstract
The two final chapters in volume I comprise a “complete theory of curves and curved surfaces”; that is, not only the “application of the differential calculus to the theory of curves” (and of curved surfaces) — what we now call differential geometry — but also the “purely algebraic part of that theory” — analytic geometry. Lacroix explained the inclusion of analytic geometry by his desire to offer a full set (“ensemble complet”) and to relate notions that were usually presented from very different points of view [Traité, I, xxv, 327].
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Belhoste [1992, 568] reads here “avec vous”; but given the teacher-pupil tone of the rest of the letter, this does not sound very convincing (unless of course Lacroix was writing that separate work as lectures for this student). Belhoste also interprets this whole passage as meaning that Lacroix intended to interpose his “descriptive geometry” [Lacroix 1795] in the Traité. I disagree: Lacroix certainly made many references to [Lacroix 1795] in chapter 5, but what he says here is that a work he had been writing on the application of analysis (to geometry, presumably) was going to be interposed in the Traité — that separate work must correspond to chapters 4 and 5.
In the 16th century the possibility of access to ancient Greek mathematical works had increased considerably because of the printing of both original versions and (usually Latin) translations. This (particularly the publication in 1588 of Commandino’s Latin translation of Pappos’ Mathematical Collection) had given origin to what Bos calls “the early modern tradition of geometrical problem solving” [Bos 2001, ch. 4].
“I choose a straight line, as AB, to which to refer all its points [i.e. those of the curve EC], and in AB I choose a point A at which to begin the investigation. [...] Then I take on the curve an arbitrary point, as C, at which we will suppose the instrument applied to describe the curve. Then I draw through C the line CB parallel to GA. Since CB and BA are unknown and indeterminate quantities, I shall call one of them y and the other x. [...] the required equation is y2 = cy − cx yb + ay − ac.” [Descartes Géométrie, 51–52]
But it should be mentioned that several of Fermat’s works, including the Isagoge, had circulated much before, in manuscript form, among the Parisian mathematicians [Boyer 1956, 82; Bos 2001, 205−206].
Quoted in [Boyer 1956, 148].
For this claim, see [Boyer 1956, 168–170].
According to [Boyer 1956, 272] its “treatment of analytic geometry is typical of the time about 1775”.
Lacroix had published a textbook Traité élémentaire de trigonométie et d’application de l’algèbre à la géométrie [1798b], combining in one volume these two subjects; [Lacroix & Bézout 1826] was a combined translation of Lacroix’s trigonometry and Bézout’s application of algebra to geometry.
[Lacroix & Bézout 1826] closes just after the study of the conic sections, so that it does not include the construction of equations. It is unlikely that this is due to the obsolescence of the subject, since an 1829 French edition of Bézout’s Cours (Paris: Bachelier) still includes that section.
This included what according to Boyer [1956, 205–206] was perhaps the first explicit appearance of the point-slope equation of the straight line: y − y′ = a(x − x′), where a is the tangent of the angle between the straight line and the abscissa axis and x′, y′ are the coordinates of a given point on it [Monge 1781, 669].
An abridged syllabus of this course is in [Langins 1987a, 130–131]. Of course, there is no guarantee that Monge really followed this syllabus. One serious possibility is that he may have taught only the geometrical applications, while others (Hachette, Malus, Dupuis) taught algebra and the calculus [Langins 1987a, 78]. See also section 8.2.
“In carefully avoiding all geometric constructions, I would have the reader realize that there exists a way of looking at geometry which one might call analytic geometry, and which consists in deducing the properties of extension from the smallest possible number of principles by purely analytic methods, as Lagrange has done in his mechanics with regard to the properties of equilibrium and movement”. This translation is taken from [Boyer 1956, 211].
Both Taton [1951, 135] and Boyer [1956, 220] wrongly ascribe this little priority to Jean-Baptiste Biot. Biot published in 1802 a Traité analytique des courbes et des surfaces du second degré; he changed the title of this work in the second edition (1805) to Essai de géométrie analytique, appliqué aux courbes et aux surfaces du second degré. Boyer had the excuse that he apparently did not see the first edition and assumed it had the same title as the second [Boyer 1956, 273]; but Taton [1951, 132] gave all these (and more) bibliographic details.
“should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation”. This translation is taken from [Boyer 1956, 212].
To be more precise, it will be divisible by tn+1, where n is the largest integer by which it would be divisible in general (that is, the multiplicity of that point). This procedure can be found in [Cramer 1750, 460–464] and [Goudin & du Séjour 1756, 77–78]. Transformation of coordinates are fundamental tools in these books.
An interesting remark is that between the curve and any of these circles it is impossible to pass another circle [l’Hôpital 1696, 73]. It is interesting because Lagrange will use this property as a definition of contact.
Of course some regularity is needed for this argument, namely that ϕ″ and F″ be bounded in a neighbourhood of x. On a different note, there is a printing error here: i2 [f″(x + j) − F″(x + j)] instead of i2 [ϕ″(x + j) − F″(x + j)]; this was later corrected (at least in the Œuvres printing [Lagrange Fonctions, 2nd ed, 187]).
Lagrange had also given these simplest osculating curves, but only as a comment, after having dealt with the general theory [Fonctions, 129–130].
It was common belief in the 18th century that all functions were piecewise monotonic. [Lagrange Fonctions, 155–156] for instance, has a similar assumption (also in a proof that the ordinate is the derivative of the area).
This characterization of envelopes can be seen for instance in [l’Hôpital 1696, ch. 8].
Henri Pitot had used the name “curves of double curvature” for space curves in 1724, but it was Clairaut [1731] who established it as standard. It was used throughout the 18th century. Neither Pitot nor Clairaut seemed to have in mind first curvature and torsion when using the name [Struik 1933, 100–101].
For each point in a curve, the radius of curvature is the radius of an evolute, but for two consecutive points in a space curve, the radii of curvature are radii of different evolutes. In fact, there is an important exception to this rule: when the curve is a line of least or greatest curvature of a surface, its centres of curvature do form an evolute. Monge implicitly reported this in [1781, 690], stating that the normals are tangent to that curve, but apparently he never recognized explicitly that it is an evolute. Lagrange [Fonctions, 183], on the other hand, was quite explicit, and Hachette cited him in a footnote in [Monge & Hachette 1799, 357].
[Tinseau 1780a, 593] has an indication of having been submitted in 1774, but according to Taton [1951, 76] the correct date is 7 December 1771. [Tinseau 1780b] has no date but, also according to Taton [1951, 76], appears to be contemporary of the former memoir.
As Taton [1951, 210] puts it, these studies take up a score (“une vingtaine”) of chapters out of about twenty-five (“quelque vingt-cinq”) in the differential part of [Monge Feuilles].
This last chapter was absent from the first edition [Taton 1951, 219].
Although Monge [Feuilles, n° 7-i] had already used it in this sense, applied to surfaces. Lagrange [Fonctions] spoke of “courbes enveloppantes” and “surfaces enveloppantes”.
Which is not correct in general. [Coolidge 1940, 136] gives the example of any non-planar curve with prime order.
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(2008). Analytic and differential geometry. In: Lacroix and the Calculus. Science Networks. Historical Studies, vol 35. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8638-2_5
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